mistake of thinking that your German pronunciation is the absolute correct pronunciation, regardless of what language is being spoken or how the original language would have pronounced it.
Well, first of all, for most languages indeed german is very close to their pronunciations (includes wild examples like Finnish, Japanese, Hungarian, Turkish ... name one). The only two examples - from my mind - where it is not is english, and according to the writing, but less to the sound: French. However you certainly find native american or south american languages that are not easy to pronounce for germans, or african for that matter ... Thai comes to mind, or Cantonese)
And second: I don't make _that_ mistake :) I can read greek, don't understand much, but can more or less fluently read a text aloud ... no greek complained so far, except for ancient rules of 'stressing' sylables.
So, to paraphrase it, the only ones who have those ridiculous claims about attic greek are americans and to a lower extend englishs.
The rest of the european world, especially the greek, pronounce it as I suggested. So, very strange if one part west of the atlantic deduces that from old songs and the other part east of it deduces the opposite. (Actually the 'american' ... and in this case only the american, not including the UK ... opinion about how latin is spoken is the same: completely ridiculous considering that we have 'reto romanic', romanian, italian, spanish, portuguese as counter examples :) )
However your 'arguing' about how Achilles is spoken, nails it: Achilles is written in greek with the letter X (chi) as the second letter. So any 'k' sound in the middle is simply wrong.
Look at the word TeX, a text setting (programming) language and how it is spoken, the 'ch' in our roman transliteration of the greek word Achilles corresponds to the X in TeX and is pronounced likewise :)
My interpretation is: 100 or more years ago a small clique of american 'scholars' _invented_ the idiotic pronunciations of latin and greek and for some reason they became 'mainstream' or 'school knowledge' in the US ... and since a few decades or even a century no one challenged it.
(I just saw an old history channel movie about the battle at the Thermopylae, it is so ashaming what nonsense the narrators are claiming ... unbelievable ... and most of them are habilitated professors in well named universities. Seems american really have trouble to simply read old sources and have the urge to reinvent/retry/reexamine half assed 'ideas' because they believe that is more accurate than an ancient text. And ofc as always the name Leonidas was a pain in the ear, but well, same for any other greek Hero)
Reminds me a bit about the scene in the movie 'Star Gate' when the main character corrects the wrong translations.
But it is always funny to meet americans on a 'latin conference' proclaiming they can 'speak' latin fluent and everyone is clapping his tights because of their 'accent' ...
Granted however: my impression is much more americans actually _speak_ latin than europeans, only the 'accent' is funny.
Btw, a friend of mine teaches classic greek, however she is a greek ... so perhaps she is 'preoccupied' :)
Sorry, no point in arguing about it. All americans have a 'wrong' school education about how classic greek is pronounced ... I did not really want to make an issue about it, sorry if you actually learned classic greek, was mot meant as an offense.